21. The normal form theorem

Normal form theorem for graphs of groups. Let be a graph of groups and .

Any can be written as
as before.

If , this expression includes `backtracking’, meaning that for some , with , and furthermore that if , then .

Similar to the case of the free group, the proof boils down to the fact that the Bass–Serre tree is a tree.

Proof. To simplify notation, set , so

.

Fix base points in the vertex spaces , which are chosen to coincide when the vertices do. Then is a loop in based at , and is a path, crossing the corresponding edge space, from to . This allows us to consider as a loop in based at . (We may assume by adding letters from a maximal tree.)

Consider the universal covering and fix a base point over in . Let be the lift of based at and its image in the Bass–Serre tree . We now analyze and closely.

Choose adjoining and so that the edge traversed by when lifted at corresponds to the coset .

Then lifts to a path in which terminates at . Similarly, lifts at to a path across the edge to the vertex space terminating at . Therefore, lifts at to a path which crosses the edge space and ends at .

Then, lifts at to a path in ending at , and lifts at to a path across the edge into the vertex space , and terminating at . Thus lifts at to a path which crosses , through , across , and ending at

.

We continue this process until we have explicitly constructed . By hypothesis, , so and are both loops in and , respectively. Since is a tree, must backtrack.

This implies that and that . That is, by Lemma 18,

.

Therefore, we have found a backtracking, and can accordingly shorten . This proves the theorem.

The lesson I got from groupoids was that it was important to move from one base point to a set of base points chosen according to the geometry of the situation – this idea was backed by Grothendieck: see my page

For the graph of groups, Higgins’ normal form theorem is a kind of `distributed computing’, in which one computer is at each base point, and from an input of a word you get a path in the graph of groups which is analysed by various computers as the path is traversed.

I think this is really pretty, and better than using trees to squash all the computers into one massive one at some arbitrary base point.

Maybe the point is also that geometric group theory deals of course with graphs and free groups, but has neglected as a tool free groupoids, i.e. the groupoid of paths in a graph.